The band structure of absorptive dielectric photonic crystals is investigated. Provided the frequency-dependent electric permeability e(c,e) satisfies certain analyticity requirements as a function of frequency, we show that no bandgaps exist in frequency regions where absorption takes place, i.e. where e(c,e) has a non-zero imaginary part. In this case real eigenvalues of the Helmholtz operator in the Bloch-decomposed formalism are absent. Using a suitable analytic continuation procedure, we find that the former change into resonances, i.e. complex numbers depending on k, the wavevector from the first Brillouin zone, thus leading to complex bands in the lower half plane. This is confirmed numerically for a simple, one-dimensional example.